multiplicatively independent

A set $X$ of nonzero complex numbers is said to be multiplicatively independent iff every equation

 $x_{1}^{\nu_{1}}x_{2}^{\nu_{2}}\cdots x_{n}^{\nu_{n}}\;=\;1$

with  $x_{1},\,x_{2},\,\ldots,\,x_{n}\in X$  and  $\nu_{1},\,\nu_{2},\,\ldots,\,\nu_{n}\in\mathbb{Z}$  implies that

 $\nu_{1}\;=\;\nu_{2}\;=\ldots=\;\nu_{n}\;=\;0.$

For example, the set of prime numbers is multiplicatively independent, by the fundamental theorem of arithmetics.

Any algebraically independent set is also multiplicatively independent.

Evidently, $\{x_{1},\,x_{2},\,\ldots,\,x_{n}\}$ is multiplicatively independent if and only if the numbers $\log x_{1}$, $\log x_{2}$, …, $\log x_{n}$ are linearly independent over $\mathbb{Q}$.  Thus the Schanuel’s conjecture may be formulated as the

Conjecture.  If  $\{x_{1},\,x_{2},\,\ldots,\,x_{n}\}$  is multiplicatively independent, then the transcendence degree of the set

 $\{x_{1},\,x_{2},\,\ldots,\,x_{n},\,\log x_{1},\,\log x_{2},\,\ldots,\,\log x_{% n}\}$

is at least $n$.

References

• 1 Diego Marques & Jonathan Sondow: Schanuel’s conjecture and algebraic powers $z^{w}$ and $w^{z}$ with $z$ and $w$ transcendental (2011). Available http://arxiv.org/pdf/1010.6216.pdfhere.
Title multiplicatively independent MultiplicativelyIndependent 2013-03-22 19:36:03 2013-03-22 19:36:03 pahio (2872) pahio (2872) 6 pahio (2872) Definition msc 11J85 msc 12F05